The problem involves understanding visual angles and how distant objects are perceived separately. This is based on basic optics and vision principles where objects need a minimum angular separation to be distinguished separately by the human eye. Here's a step-by-step solution:
The key concept here is that for a person to distinguish between two objects (in this case, the two pillars), the angle subtended by the objects at the observer's eye should be at least the minimum angular resolution of the human eye. This is approximately \(1/60\) of a degree, which translates to about \(1\) minute of arc.
The situation can be understood in terms of a triangle where the observer is at one vertex and the two pillars are at the other vertices. We need to find the base of this triangle (the distance between the two pillars), given the height (the distance from the observer to the midpoint between the pillars, which is \(11 \text{ km} = 11000 \text{ m}\)).
For small angles, we can use the approximation: \(\theta \approx \frac{d}{D}\), where \(\theta\) is the angle in radians, \(d\) is the distance between the objects, and \(D\) is the distance from the observer.
We know that \(\theta \approx \frac{1}{60} \times \frac{\pi}{180} \text{ radians}\) because the angle is converted from degrees to radians.
Thus, the formula becomes: \(\frac{1}{60} \times \frac{\pi}{180} \approx \frac{d}{11000}\).
Rounding \(3.1988 \text{ meters}\) gives approximately \(3 \text{ meters}\).
Therefore, the approximate distance between the two pillars that allows the observer to see them separately is \(3 \, \text{meters}\). Hence, the correct option is 3 m.